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Atsushi Shiho

Publications and source records attributed to Atsushi Shiho.

At least 19 recordsLinked to original sources
arXiv2025

Theory of weights for log convergent cohomologies I: the case of a proper smooth scheme with an SNCD in characteristic p>0

Using log convergent topoi, %In the derived category of filtered complexes of %sheaves of modules over %an isostructure we define two fundamental filtered complexes $(E_{conv},P)$ and $(C_{conv},P)$ for the log scheme obtained by a smooth scheme with a relative simple normal crossing divisor over a scheme of characteristic $p>0$. Using $(C_{conv},P)$, we prove the $p$-adic purity. As a corollary of it, we prove that $(E_{conv},P)$ and $(C_{conv},P)$ are canonically isomorphic. These filtered complexes produce the weight spectral sequence of the log convergent cohomology sheaf of the log scheme. We also give the comparison theorem between the projections of $(E_{conv},P)$ and $(C_{conv},P)$ to the derived category of bounded below filtered complexes of sheaves of modules in the Zariski topos of the log scheme and the weight-filtered isozariskian filtered complex $(E_{zar},P)_{Q}$ of the log scheme defined in our previous book.

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arXiv2025

Integral p-adic cohomology theories for open and singular varieties

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically étale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.

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arXiv2024

Blow-up invariance for Hodge-Witt sheaves with modulus

In this paper, we prove the blow-up invariance for Hodge-Witt sheaves with modulus, which is a generalization of a result of Koizumi for Witt sheaves and that of Kelly-Miyazaki and Koizumi for Hodge sheaves. As a consequence, we obtain the representability of Hodge-Witt sheaves with modulus in the category of motives with modulus under the assumption of resolution of singularities.

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arXiv2020

Comparison of relatively unipotent log de Rham fundamental groups

In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta-Iovita-Kim's article: obtaining in this way a complete algebraic criterion for good reduction for curves.

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arXiv2018

On comparison between relative log de Rham-Witt cohomology and relative log crystalline cohomology

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.

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arXiv2016

Convergent isocrystals on simply connected varieties

It is conjectured by de Jong that, if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial étale fundamental group, any isocrystal on on $X/W$ is trivial. We prove this conjecture under two additional assumptions. Version 2: the main change is an addendum. We prove that if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial étale fundamental group, any infinitesimal isocrystal on $X/W$ is trivial. To this aim we wrote some general facts on such infinitesimal isocrystals over W which are missing in the literature.

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arXiv2015

Chern classes of crystals

The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong's conjecture relating the geometric étale fundamental group of a smooth projective variety defined over a perfect field and the triviality of its category of isocrystals. We also discuss the case of the Gauß-Manin convergent isocrystal.

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arXiv2014

A note on convergent isocrystals on simply connected varieties

It is conjectured by de Jong that, if $X$ is a connected projective smooth variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial etale fundamental group, any convergent isocrystal $\mathcal{E}$ on $X$ is trivial. We discuss this conjecture when $X$ is liftable to characteristic zero, and prove the triviality of $\mathcal{E}$ in this case under certain conditions on (semi)stability.

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arXiv2014

On $p$-adic differential equations on semistable varieties II

This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcategories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories.

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arXiv2012

Parabolic log convergent isocrystals

In this paper, we introduce the notion of parabolic log convergent isocrystals on smooth varieties endowed with a simple normal crossing divisor, which is a kind of $p$-adic analogue of the notion of parabolic bundles on smooth varieties defined by Seshadri, Maruyama-Yokogawa, Iyer-Simpson, Borne. We prove that the equivalence between the category of $p$-adic representations of the fundamental group and the category of unit-root convergent $F$-isocrystals (proven by Crew) induces the equivalence between the category of $p$-adic representations of the tame fundamental group and the category of unit-root semisimply adjusted parabolic log convergent $F$-isocrystals. We also prove equivalences which relate categories of log convergent isocrystals on certain fine log algebraic stacks with some conditions and categories of adjusted parabolic log convergent isocrystals with some conditions. We also give an interpretation of unit-rootness in terms of the generic semistability with slope 0. Our result can be regarded as a $p$-adic analogue of the results of Seshadri, Mehta-Seshadri, Iyer-Simpson and Borne.

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arXiv2012

Notes on generalizations of local Ogus-Vologodsky correspondence

Given a smooth scheme over $\Z/p^n\Z$ with a lift of relative Frobenius to $\Z/p^{n+1}\Z$, we construct a functor from the category of Higgs modules to that of modules with integrable connections as the composite of the level raising inverse image functors from the category of modules with integrable $p^{m}$-connections to that of modules with integrable $p^{m-1}$-connections for $1 \leq m \leq n$. In the case $m=1$, we prove that the level raising inverse image functor is an equivalence when restricted to quasi-nilpotent objects, which generalizes a local result of Ogus-Vologodsky. We also prove that the above level raising inverse image functor for a smooth $p$-adic formal scheme induces an equivalence of $\Q$-linearized categories for general $m$ when restricted to nilpotent objects (in strong sense), under a strong condition on Frobenius lift. We also prove a similar result for the category of modules with integrable $p^{m}$-Witt-connections.

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arXiv2010

Purity for overconvergence

Let $X \hookrightarrow \overline{X}$ be an open immersion of smooth varieties over a field of characteristic $p>0$ such that the complement is a simple normal crossing divisor and let $\overline{Z} \subseteq Z \subseteq \overline{X}$ be closed subschemes of codimension at least $2$. In this paper, we prove that the canonical restriction functor between the category of overconvergent $F$-isocrystals $F\text{-}{\rm Isoc}^{\dagger}(X,\overline{X}) \longrightarrow F\text{-}{\rm Isoc}^{\dagger}(X \setminus Z, \overline{X} \setminus \overline{Z})$ is an equivalence of categories. We also prove an application to the category of $p$-adic representations of the fundamental group of $X$, which is a higher-dimensional version of a result of Tsuzuki.

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arXiv2009

Cut-by-curves criterion for the log extendability of overconvergent isocrystals

In this paper, we prove a `cut-by-curves criterion' for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor, under certain assumption. This is a $p$-adic analogue of a version of cut-by-curves criterion for regular singuarity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of $\nabla$-modules.

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arXiv2009

On logarithmic extension of overconvergent isocrystals

In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a $p$-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0.

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arXiv2008

Relative log convergent cohomology and relative rigid cohomology I

In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically.

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